# Article

Full entry | PDF   (0.2 MB)
Keywords:
block designs; orbits; projective linear group; projective special linear group; twisted projective linear group; Kramer-Mesner method
Summary:
Using the Kramer-Mesner method, $4$-$(26,6,\lambda)$ designs with $PSL(2,25)$ as a group of automorphisms and with $\lambda$ in the set $\{30,51,60,81,90,111\}$ are constructed. The search uses specific partitioning of columns of the orbit incidence matrix, related to so-called quasi-designs''. Actions of groups $PSL(2,25)$, $PGL(2,25)$ and twisted $PGL(2,25)$ are being compared. It is shown that there exist $4$-$(26,6,\lambda)$ designs with $PGL(2,25)$, respectively twisted $PGL(2,25)$ as a group of automorphisms and with $\lambda$ in the set $\{51,60,81,90,111\}$. With $\lambda$ in the set $\{60,81\}$, there exist designs which possess all three considered groups as groups of automorphisms. An overview of $t$-$(q+1,k,\lambda)$ designs with $PSL(2,q)$ as group of automorphisms and with $(t,k) \in \{(4,5), (4,6), (5,6)\}$ is included.
References:
[1] Acketa D.M., Mudrinski V.: A $4$-design on $38$ points. submitted.
[2] Acketa D.M., Mudrinski V., Paunić Dj.: A search for $4$-designs arising by action of $PGL(2,q)$. Publ. Elektrotehn. Fak., Univ. Beograd, Ser. Mat. 5 (1994), 13-18. MR 1322263 | Zbl 0816.05017
[3] Acketa D.M., Mudrinski V.: Two $5$-designs on $32$ points. accepted for Discrete Mathematics. Zbl 0873.05010
[4] Alltop W.O.: An infinite class of $4$-designs. J. Comb. Th. 6 (1969), 320-322. MR 0241316 | Zbl 0169.01903
[5] Beth T., Jungnickel D., Lenz B.: Design Theory. Bibliographisches Institut Mannheim- Wien-Zürich, 1985. MR 0779284 | Zbl 0945.05005
[6] Chee Y.M., Colbourn C.J., Kreher D.L.: Simple $t$-designs with $t \leq 30$. Ars Combinatoria 29 (1990), 193-258. MR 1046108
[7] Dautović S., Acketa D.M., Mudrinski V.: A graph approach to isomorphism testing of $4$-$(48,5,\lambda)$ designs arising from $PSL(2,47)$. submitted.
[8] Denniston R.H.F.: Some new $5$-designs. Bull. London Math. Soc. 8 (1976), 263-267. MR 0480077 | Zbl 0339.05019
[9] Driessen L.M.H.E.: $t$-designs, $t \geq 3$. Tech. Report (1978), Department of Mathematics, Eindhover University of Technology, Holland.
[10] Gorenstein D.: Finite Simple Groups, An Introduction to Their Classification. Plenum Press, New York, London, 1982. MR 0698782 | Zbl 0672.20010
[11] Huppert B.: Endliche Gruppen, I. Die Grundlehren der matematischen Wissenschaften. Band 134 (1967), Springer-Verlag, Berlin, Heidelberg, New York, xii + 793 pp. MR 0224703
[12] Huppert B., Blackburn N.: Finite Groups, III. Die Grundlehren der matematischen Wissenschaften, Band 243 (1982), Springer-Verlag, Berlin, Heidelberg, New York, p.454. MR 0662826 | Zbl 0514.20002
[13] Janko Z., Tonchev V.: Private communication.
[14] Kramer E.S., Leavitt D.W., Magliveras S.S.: Construction procedures for $t$-designs and the existence of new simple $6$-designs. Ann. Discrete Math. 26 (1985), 247-274. MR 0833794 | Zbl 0585.05002
[15] Kramer E.S., Mesner D.M.: $t$-designs on hypergraphs. Discrete Math. 15 (1976), 263-296. MR 0460143 | Zbl 0362.05049
[16] Kreher D.L., Radziszowski S.P.: The existence of simple $6$-$(14,7,4)$ designs. Jour. of Combinatorial Theory, Ser. A 43 (1986), 237-243. MR 0867649 | Zbl 0647.05013
[17] Kreher D.L., Radziszowski S.P.: Simple $5$-$(28,6,\lambda)$ designs from $PSL_2(27)$. Ann. Discrete Math. 34 (1987), 315-318. MR 0920656

Partner of